Backpacking is a relatively slow endeavor. To complete the trip you'd need to hike almost 17 miles a day, without taking a day off. Hiking on roads, which are relatively flat and even would be faster than parts of the Appalachian Trail, but if you aren't used to that kind of hiking you would find it very difficult. Most people would probably average 10-12 miles per day. Going around Atlanta could add quite a few miles as well, I'm guessing you'd need to walk closer to 20 miles a day.
Finding places to camp would be difficult. There is a lot of distance between most public parks and not nearly enough for you to sleep at one every night. Sleeping on someones land without permission can get you in trouble most places.
It's a neat idea, but it would be physically demanding to accomplish in one month. The logistics would also hard, finding places to stay. Don't forget about the cost, while transportation (shoes) and lodging (outside) would be cheap, you'd still need to get food. All that walking means you need to eat more and it wouldn't be hard to eat several hundred dollars worth.
